PORTUGALIAE MATHEMATICA
Vol. 58 Fasc. 3 – 2001
Nova Série
VARIATIONAL AND TOPOLOGICAL METHODS
FOR DIRICHLET PROBLEMS
WITH p-LAPLACIAN
G. Dinca, P. Jebelean and J. Mawhin
Presented by L. Sanchez
0 – Introduction
The aim of this paper is to obtain existence results for the Dirichlet problem
(
−∆p u = f (x, u)
in Ω ,
u|∂Ω = 0 .
∂
∂u
Here ∆p = ∂x
(|∇u|p−2 ∂x
), 1 < p < ∞, is the so-called p-Laplacian and
i
i
f : Ω×R → R is a Carathéodory function which satisfies some special growth
conditions. One of the main ideas is to present the operator −∆p as a duality
0
mapping between W01,p (Ω) and its dual W −1,p (Ω), p1 + p10 = 1, corresponding
to the normalization function ϕ(t) = tp−1 . This idea, coming from Lions’ book
[23], proves to be a very fruitful one. The properties of the Nemytskii operator
(Nf u)(x) = f (x, u(x)), generated by the Carathéodory function f , the homotopy invariance of the Leray–Schauder degree (under the form of a priori estimate, uniformly with respect to λ ∈ [0, 1], of the solutions set of the equation
u = λ(−∆p )−1 Nf u with (−∆p )−1 Nf : W01,p (Ω) → W01,p (Ω) compact), the well
known Mountain Pass Theorem of Ambrosetti and Rabinowitz and the variational characterization of the first eigenvalue of −∆p on W01,p (Ω) are the other
essential tools which are also used.
Received : November 24, 2000.
340
G. DINCA, P. JEBELEAN and J. MAWHIN
1 – The p-Laplacian as duality mapping
The main idea of this paragraph is to present the operator −∆p , 1 < p < ∞,
0
as duality mapping Jϕ : W01,p (Ω) → W −1,p (Ω), p1 + p10 = 1, corresponding to the
normalization function ϕ(t) = tp−1 .
Originated in the well known book of Lions (see [23]), this presentation has
the advantage of allowing to apply the general results known for the duality
mapping to the particular case of the p-Laplacian. For example, the surjectivity
of the duality mapping (itself an immediate consequence of a well known result
of Browder (see e.g. [8])) achieves the existence of the W01,p (Ω)-solution for the
0
0
equation −∆p u = f , with f ∈ W −1,p (Ω). Note that if f ∈ W −1,p (Ω) is given,
then an element u ∈ W01,p (Ω) is said to be solution of the Dirichlet problem
(
−∆p u = f
in Ω ,
u|∂Ω = 0 ,
0
if the equality −∆p u = f is satisfied in the sense of W −1,p (Ω).
For the convenience of the reader we have considered to put away the definitions and the results concerning the duality mapping, which will be used in
the sequel. Because these results are already known, the proof is often omitted; however the proof is given when these results achieve specific properties for
p-Laplacian.
1.1. Basic results concerning the duality mapping
Below, X always is a real Banach space, X ∗ stands for its dual and h·, ·i is
the duality between X ∗ and X. The norm on X and on X ∗ is denoted by k k.
Given a set valued operator A : X → P(X ∗ ), the range of A is defined to be
the set
R(A) =
[
Ax
x∈D(A)
where D(A) = {x ∈ X | Ax 6= ∅} is the domain of A. The operator A is said to
be monotone if
hx∗1 − x∗2 , x1 − x2 i ≥ 0
whenever x1 , x2 ∈ D(A) and x∗1 ∈ Ax1 , x∗2 ∈ Ax2 .
341
DIRICHLET PROBLEMS WITH p-LAPLACIAN
A continuous function ϕ : R+ → R+ is called a normalization function if it is
strictly increasing, ϕ(0) = 0 and ϕ(r) → ∞ with r → ∞.
By duality mapping corresponding to the normalization function ϕ, we mean
the set valued operator Jϕ : X → P(X ∗ ) as following defined
Jϕ x =
n
x∗ ∈ X ∗ | hx∗ , xi = ϕ(kxk) kxk, kx∗ k = ϕ(kxk)
o
for x ∈ X.
By the Hahn–Banach theorem, it is easy to check that D(Jϕ ) = X.
Some of the main properties of the duality mapping are contained in the
following
Theorem 1. If ϕ is a normalization function, then:
(i) for each x ∈ X, Jϕ x is a bounded, closed and convex subset of X ∗ ;
(ii) Jϕ is monotone:
³
hx∗1 − x∗2 , x1 − x2 i ≥ ϕ(kx1 k) − ϕ(kx2 k)
´³
´
kx1 k − kx2 k ≥ 0 ,
for each x1 , x2 ∈ X and x∗1 ∈ Jϕ x1 , x∗2 ∈ Jϕ x2 ;
(iii) for each x ∈ X, Jϕ x = ∂ψ(x), where ψ(x) =
kxk
R
ϕ(t) dt and ∂ψ : X →
0
P(X ∗ ) is the subdifferential of ψ in the sense of convex analysis, i.e.
∂ψ(x) =
n
x∗ ∈ X ∗ | ψ(y) − ψ(x) ≥ hx∗ , y − xi for all y ∈ X
o
.
For proof we refer to Beurling and Livingston [5], Browder [8], Lions [23],
Ciorãnescu [9].
Remark 1. We recall that a functional f : X → R is said to be Gâteaux
differentiable at x ∈ X if there exists f 0 (x) ∈ X ∗ such that
f (x + t h) − f (x)
= hf 0 (x), hi
t→0
t
lim
for all h ∈ X.
If the convex function f : X → R is Gâteaux differentiable at x ∈ X, then
it is a simple matter to verify that ∂f (x) consists of a single element, namely
x∗ = f 0 (x).
This simple remark will be essentially used in the sequel.
342
G. DINCA, P. JEBELEAN and J. MAWHIN
The geometry of the space X (or X ∗ ) supplies further properties of the duality
mapping. That is why we recall the following (see e.g. Diestel [11])
Definition 1. The space X is said to be:
(a) uniformly convex if for each ε ∈ (0, 2], there exists δ(ε) > 0 such that
if kxk = kyk = 1 and kx − yk ≥ ε then kx + yk ≤ 2 (1 − δ(ε));
(b) locally uniformly convex if from kxk = kxn k = 1 and kxn + xk → 2 with
n → ∞, it results that xn → x (strongly in X);
(c) strictly convex if for each x, y ∈ X with kxk = kyk = 1, x 6= y and
λ ∈ (0, 1), we have kλ x + (1−λ) yk < 1.
Theorem 2. The following implications hold:
X uniformly convex =⇒ X locally uniformly convex =⇒ X strictly convex .
For proof we refer to Diestel [11].
Theorem 3 (Pettis–Milman). If X is uniformly convex then X is reflexive.
For proof see e.g. Brézis [6] or Diestel [11] — where the original proof of Pettis
is given.
In the sequel, ϕ will be a normalization function.
Proposition 1.
(i) If X is strictly convex, then Jϕ is strictly monotone:
hx∗1 − x∗2 , x1 − x2 i > 0
for each x1 , x2 ∈ X, x1 6= x2 and x∗1 ∈ Jϕ x1 , x∗2 ∈ Jϕ x2 ; in particular,
Jϕ x1 ∩ Jϕ x2 = φ if x1 6= x2 .
(ii) If X ∗ is strictly convex, then card(Jϕ x) = 1, for all x ∈ X.
Proof: (i) First, it is easy to check that (see e.g. James [18]) if X is strictly
convex, then for each x∗ ∈ X ∗ \{0} there exists at most an element x ∈ X with
kxk = 1, such that hx∗ , xi = kx∗ k.
Now, supposing by contradiction that there exist x1 , x2 ∈ X with x1 6= x2
and x∗1 ∈ Jϕ x1 , x∗2 ∈ Jϕ x2 , satisfying
hx∗1 − x∗2 , x1 − x2 i = 0
343
DIRICHLET PROBLEMS WITH p-LAPLACIAN
we have
³
0 = hx∗1 − x∗2 , x1 − x2 i ≥ ϕ(kx1 k) − ϕ(kx2 k)
´³
´
kx1 k − kx2 k ≥ 0
and so, we get kx1 k = kx2 k.
Remark that x1 6= x2 , kx1 k = kx2 k implies x1 6= 0, x2 6= 0.
We obtain
0 =
¿
x∗1
−
x∗2 ,
"
x1
x2
−
kx1 k kx2 k
¿
x2
= ϕ(kx1 k) − x∗1 ,
kx2 k
À
À#
"
¿
x1
+ ϕ(kx2 k) − x∗2 ,
kx1 k
À#
and both of the brackets being positive, it results
kx∗1 k = ϕ(kx1 k) =
which together with
kx∗1 k
yields
¿
x∗1 ,
x2
kx2 k
À
=
¿
x∗1 ,
¿
x∗1 ,
x1
kx1 k
= kx∗1 k =
x2
kx2 k
À
À
¿
x∗1 ,
x1
kx1 k
By the above mentioned result of James, we have
which is a contradiction.
À
.
x1
kx1 k
=
x2
kx2 k
i.e. x1 = x2
(ii) It results from the fact that Jϕ x is a convex part of ∂B(0, ϕ(kxk)) =
{x∗ ∈ X ∗ | kx∗ k = ϕ(kxk)}.
Proposition 2. If X is locally uniformly convex and Jϕ is single valued
(Jϕ : X → X ∗ ), then Jϕ satisfies the (S+ ) condition: if xn * x (weakly in X)
and lim sup hJϕ xn , xn − xi ≤ 0 then xn → x (strongly in X).
n→∞
Proof: It is immediately that from xn * x and lim sup hJϕ xn , xn − xi ≤ 0
n→∞
it results that lim sup hJϕ xn − Jϕ x, xn − xi ≤ 0.
n→∞
By
³
0 ≤ ϕ(kxn k) − ϕ(kxk)
´³
´
kxn k − kxk ≤
D
Jϕ xn − Jϕ x, xn − x
E
344
it follows
G. DINCA, P. JEBELEAN and J. MAWHIN
³
ϕ(kxn k) − ϕ(kxk)
´³
´
kxn k − kxk| → 0
and hence, kxn k → kxk.
Now, by a well known result (see e.g. Diestel [11]), in a locally uniformly
convex space, from xn * x and kxn k → kxk it results xn → x.
Proposition 3. If X is reflexive and Jϕ : X → X ∗ then Jϕ is demicontinuous:
if xn → x in X then Jϕ xn * Jϕ x in X ∗ .
Proof: By the boundedness of (xn ) it follows that (Jϕ xn ) is bounded in X ∗ .
Since X ∗ is also reflexive, in order to prove that Jϕ xn * Jϕ x it suffices to show
that all subsequences of (Jϕ xn ) which are weakly convergent have the same limit,
namely Jϕ x.
Let x∗ ∈ X ∗ be the weak limit of a subsequence of (Jϕ xn ), still denoted by
(Jϕ xn ).
By the weakly lower semicontinuity of the norm, we have:
kx∗ k ≤ lim inf kJϕ xn k = lim ϕ(kxn k) = ϕ(kxk) .
n→∞
n→∞
On the other hand, from xn → x and Jϕ xn * x∗ , it follows that
hJϕ xn , xn i → hx∗ , xi .
But,
hJϕ xn , xn i = ϕ(kxn k) kxn k → ϕ(kxk) kxk .
We get hx∗ , xi = ϕ(kxk) kxk and so, ϕ(kxk) ≤ kx∗ k. Finally hx∗ , xi = ϕ(kxk) kxk
and ϕ(kxk) = kx∗ k, which means x∗ = Jϕ x.
Theorem 4. Let X be reflexive and Jϕ : X → X ∗ . Then R(Jϕ ) = X ∗ .
Proof: The result follows from a well known theorem of Browder [8]: if X
is reflexive and T : X → X ∗ is monotone, hemicontinuous and coercive, then T
is surjective.
In our case, Jϕ is monotone by Theorem 1 (ii). The fact that Jϕ is hemicontinuous means:
D
E
lim Jϕ (u + t v), w = hJϕ u, wi
t→0
for u, v, w ∈ X, and it results from the demicontinuity of Jϕ (Proposition 3).
345
DIRICHLET PROBLEMS WITH p-LAPLACIAN
Finally,
hJϕ u, ui
= ϕ(kuk) → ∞
kuk
with kuk → ∞ ,
hence Jϕ is coercive.
Theorem 5. Let X be reflexive, locally uniformly convex and Jϕ : X → X ∗ .
Then Jϕ is bijective with its inverse Jϕ−1 bounded, continuous and monotone.
Moreover, it holds
Jϕ−1 = χ−1 Jϕ∗−1
where χ : X → X ∗∗ , is the canonical isomorphism between X and X ∗∗ and Jϕ∗−1 :
X ∗ → X ∗∗ is the duality mapping on X ∗ corresponding to the normalization
function ϕ−1 .
Proof: By Theorem 4, Jϕ is surjective. The space X being locally uniformly
convex, it is strictly convex (Theorem 2) and by Proposition 1 (i) we have that
Jϕ is injective.
Let, now, χ be the canonical isomorphism between X and X ∗∗ (hχ(x), x∗ i =
hx∗ , xi) and let Jϕ∗−1 : X ∗ → X ∗∗ be the duality mapping corresponding to the
normalization function ϕ−1 . It should be noticed that because X is reflexive and
locally uniformly convex, so is X ∗∗ ; in particular X ∗∗ is strictly convex (Theorem 2) and, consequently, Jϕ∗−1 : X ∗ → X ∗∗ is single valued (Proposition 1 (ii)).
It is easy to see that:
Jϕ−1 = χ−1 Jϕ∗−1 .
(1)
From (1) and because a duality mapping maps bounded subsets into bounded
subsets, it is immediately that Jϕ−1 is bounded.
To see that Jϕ−1 is continuous, let x∗n → x∗ in X ∗ .
From (1) and by Proposition 3 we have that Jϕ−1 x∗n * Jϕ−1 x∗ . By the definition of the duality mapping Jϕ , it is easy to see that kJϕ−1 x∗n k → kJϕ−1 x∗ k. But
the space X is assumed to be locally uniformly convex, and so, Jϕ−1 x∗n → Jϕ−1 x∗ .
To prove the monotonicity of Jϕ−1 , the space X is identified with X ∗∗ by the
canonical isomorphism χ. Then, for x∗1 , x∗2 ∈ X ∗ , we have:
D
χ(Jϕ−1 x∗1 ) − χ(Jϕ−1 x∗2 ), x∗1 − x∗2
E
=
D
Jϕ∗−1 x∗1 − Jϕ∗−1 x∗2 , x∗1 − x∗2
E
and we apply Theorem 1 (ii) with ϕ−1 instead of ϕ and X ∗ instead of X.
346
G. DINCA, P. JEBELEAN and J. MAWHIN
1.2. The functional framework
In the sequel, Ω will be a bounded domain in RN , N ≥ 2 with Lipschitz
continuous boundary and p ∈ (1, ∞). We shall use the standard notations:
W
1,p
½
∂u
u ∈ L (Ω) |
∈ Lp (Ω), i = 1, ..., N
∂xi
(Ω) =
p
¾
equipped with the norm:
kukpW 1,p (Ω)
kukp0,p
=
°
N °
X
° ∂u °p
°
°
+
° ∂x °
i 0,p
i=1
where k k0,p is the usual norm on Lp (Ω).
It is well known that (W 1,p (Ω), k kW 1,p (Ω) ) is separable, reflexive and uniformly convex (see e.g. Adams [1, Theorem 3.5]).
We need the space
W01,p (Ω) = the closure of C0∞ (Ω) in the space W 1,p (Ω)
=
n
u ∈ W 1,p (Ω) | u|∂Ω = 0
o
the value of u on ∂Ω being understood in the sense of the trace: there is a
1− 1 ,p
unique linear and continuous operator γ : W 1,p (Ω) → W p (∂Ω) such that
γ is surjective and for u ∈ W 1,p (Ω) ∩ C(Ω) we have γ u = u|∂Ω . It holds
W01,p (Ω) = ker γ.
0
The dual space (W01,p (Ω))∗ will be denoted by W −1,p (Ω), where p1 + p10 = 1.
For each u ∈ W 1,p (Ω), we put
∇u =
µ
∂u
∂u
, ...,
∂x1
∂xN
¶
,
|∇u| =
Ã
¶
N µ
X
∂u 2
i=1
∂xi
!1
2
and let us remark that
|∇u| ∈ Lp (Ω) ,
|∇u|p−2
∂u
0
∈ Lp (Ω)
∂xi
for i = 1, ..., N .
0
Therefore, by the theorem concerning the form of the elements of W −1,p (Ω) (see
Brézis [6] or Lions [23]) it follows that the operator −∆p may be seen acting from
0
W01,p (Ω) into W −1,p (Ω) by
h−∆p u, vi =
Z
Ω
|∇u|p−2 ∇u ∇v
for u, v ∈ W01,p (Ω) .
347
DIRICHLET PROBLEMS WITH p-LAPLACIAN
By virtue of the Poincaré inequality
kuk0,p ≤ Const(Ω, n) k|∇u|k0,p
for all u ∈ W01,p (Ω) ,
the functional
W01,p (Ω) 3 u 7−→ kuk1,p : = k|∇u|k0,p
is a norm on W01,p (Ω), equivalent with k kW 1,p (Ω) .
Because the geometrical properties of the space are not automatically maintained by passing to an equivalent norm, we give a direct proof of the following
theorem
Theorem 6. The space (W01,p (Ω), k k1,p ) is uniformly convex.
Proof: First, let p ∈ [2, ∞). Then (see e.g. Adams [1, pp. 36]) for each
z, w ∈ RN , it holds:
¯
¯
¯p
¯
³
´
¯ z + w ¯p
¯
¯
¯
¯ + ¯ z − w ¯ ≤ 1 |z|p + |w|p .
¯ 2 ¯
¯ 2 ¯
2
Let u, v ∈ W01,p satisfy kuk1,p = kvk1,p = 1 and ku − vk1,p ≥ ε ∈ (0, 2]. We
have
°
°
°p
°
° u + v °p
°
°
°
° + °u − v °
° 2 °
° 2 °
1,p
1,p
¯
¯ !
¯
Z ï
¯ ∇u + ∇v ¯p ¯ ∇u − ∇v ¯p
¯
¯
¯
¯
=
¯ +¯
¯
¯
2
2
Ω
1
≤
2
Z ³
|∇u|p + |∇v|p
Ω
´
=
´
1³
kukp1,p + kvkp1,p = 1
2
which yields
(2)
°
°
° u + v °p
°
°
° 2 °
1,p
≤ 1−
µ ¶p
ε
2
.
If p ∈ (1, 2), then (see e.g. Adams [1, pp. 36]) for each z, w ∈ RN , it holds:
1
¯ 0 ¯
¯ 0
¯
· ³
´¸ p−1
¯ z + w ¯p ¯ z − w ¯p
1
p
p
¯
¯
¯
¯
.
¯ 2 ¯ + ¯ 2 ¯ ≤ 2 |z| + |w|
A straightforward computation shows that if v ∈ W01,p (Ω) then |∇v|p ∈ Lp−1 (Ω)
0
0
and k|∇v|p k0,p−1 = kvkp1,p .
0
348
G. DINCA, P. JEBELEAN and J. MAWHIN
Let v1 , v2 ∈ W01,p (Ω). Then |∇v1 |p , |∇v2 |p ∈ Lp−1 (Ω), with 0 < p − 1 < 1
and, according to Adams [1, pp. 25],
0
°
°
0
0°
°
°|∇v1 |p + |∇v2 |p °
0
0
0
≥ k|∇v1 |p k0,p−1 + k|∇v2 |p k0,p−1 .
0,p−1
Consequently,
°
° 0
° p0
¯ 0°
¯ 0°
°
°¯
°¯
° v1 + v 2 ° p
°
°
° ¯ v1 + v 2 ¯ p °
° ¯ v1 − v 2 ¯ p °
°
° + ° v1 − v 2 °
¯
¯ °
°
°
°
¯
¯
= ° ¯∇
+ ∇
°
2 °1,p ° 2 °1,p
2 ¯ °0,p−1 °¯
2 ¯ °0,p−1
°
°¯
°¯ v + v ¯¯p0 ¯¯ v − v ¯¯p0 °
1
2¯ °
1
2¯
°¯
¯
+ ¯∇
≤ ° ¯∇
°
°
2 ¯
2 ¯ °
0,p−1

 1
¯ 0 ¯
¯ 0 !p−1 p−1
Z ï
¯ ∇v1 +∇v2 ¯p ¯ ∇v1 −∇v2 ¯p
¯
¯ +¯
¯

= 
¯
¯
¯
¯
2
2
Ω
≤
"
1
2
=
·
1
1
kv1 kp1,p + kv2 kp1,p
2
2
Z ³
p
|∇v1 | + |∇v2 |
p
Ω
´
¸
#
1
p−1
1
p−1
.
For u, v ∈ W01,p (Ω) with kuk1,p = kvk1,p = 1 and ku − vk1,p ≥ ε ∈ (0, 2], we
get
(3)
°
° 0
µ ¶p0
° u + v °p
ε
°
°
.
≤
1
−
° 2 °
2
1,p
From (2) and (3), in either case there exists δ(ε) > 0 such that ku + vk 1,p ≤
2 (1 − δ(ε)).
Below, the space W01,p (Ω) always will be considered to be endowed with the
norm k k1,p .
Theorem 7. The operator −∆p : W01,p (Ω) → W −1,p (Ω) is a potential one.
More precisely, its potential is the functional ψ : W01,p (Ω) → R, given by
0
ψ(u) =
1
kukp1,p
p
and
ψ 0 = −∆p = Jϕ
where Jϕ : W01,p (Ω) → W −1,p (Ω) is the duality mapping corresponding to the
normalization function ϕ(t) = tp−1 .
0
DIRICHLET PROBLEMS WITH p-LAPLACIAN
Proof: Since ψ(u) =
349
kuk
R 1,p
ϕ(t) dt, it is sufficient to prove that ψ is Gâteaux
0
differentiable and ψ 0 (u) = −∆p u for all u ∈ W01,p (Ω) (see Theorem 1 (iii) and
Remark 1).
If u ∈ W01,p (Ω) is such that |∇u| = 0Lp (Ω) (this implies that kuk1,p = 0 i.e.
u = 0W 1,p (Ω) ), then it is immediately that hψ 0 (u), hi = 0 for all h ∈ W01,p (Ω).
0
Therefore, we may suppose that |∇u| 6= 0Lp (Ω) .
It is obvious that ψ can be written as a product ψ = QP , where Q : Lp (Ω) → R
is given by Q(v) = p1 kvkp0,p and P : W01,p (Ω) → Lp (Ω) is given by P (v) = |∇v|.
The functional Q is Gâteaux differentiable (see Vainberg [28]) and
hQ0 (v), hi = h|v|p−1 sign v, hi
(4)
for all v, h ∈ Lp (Ω).
Simple computations show that the operator P is Gâteaux differentiable at u
and
∇u ∇v
(5)
P 0 (u) · v =
|∇u|
for all v ∈ W01,p (Ω).
Combining (4) and (5), we obtain that ψ is Gâteaux differentiable at u and
hψ 0 (u), vi =
=
=
D
Q0 (P (u)), P 0 (u) · v
¿
Z
|∇u|
p−1
∇u ∇v
,
|∇u|
À
E
|∇u|p−2 ∇u ∇v = h−∆p u, vi
Ω
for all v ∈ W01,p (Ω).
Remark 2. Let k k∗ be the dual norm of k k1,p . Then, we have
p−1
k− ∆p uk∗ = kJϕ uk∗ = ϕ(kuk1,p ) = kuk1,p
.
Theorem 8. The operator −∆p defines a one-to-one correspondence be0
tween W01,p (Ω) and W −1,p (Ω), with inverse (−∆p )−1 monotone, bounded and
continuous.
Proof: It is obvious from Theorems 7, 6 and 5.
350
G. DINCA, P. JEBELEAN and J. MAWHIN
0
Remark 3. In fact, the above Theorem 8 asserts that for each f ∈ W −1,p (Ω),
the equation −∆p u = f has a unique solution in W01,p (Ω).
The properties of (−∆p )−1 show how the solution u = (−∆p )−1 f depends on
the data f . These properties will be used in the sequel.
Since the elements of W01,p (Ω) vanish on the boundary ∂Ω in the sense of the
trace, it is natural that the unique solution in W01,p (Ω) of the equation −∆p u = f
to be called solution of the Dirichlet problem
(
−∆p u = f ,
u|∂Ω = 0 .
We shall conclude this section with two technical results which will be useful
in the sequel.
We have seen (Theorem 7) that the functional ψ(u) = p1 kukp1,p is Gâteaux
differentiable on W01,p (Ω). Moreover, we have:
Theorem 9.
The functional ψ is continuously Fréchet differentiable on
W01,p (Ω).
For the proof we need the following lemma (see Glowinski and Marrocco [16]).
Lemma 1.
(i) If p ∈ [2, ∞) then it holds:
¯
¯
³
´p−2
¯
¯ p−2
z − |y|p−2 y ¯ ≤ β |z − y| |z| + |y|
¯|z|
for all y, z ∈ RN
with β independent of y and z;
(ii) If p ∈ (1, 2], then it holds:
¯
¯
¯
¯ p−2
z − |y|p−2 y ¯ ≤ β |z − y|p−1
¯|z|
for all y, z ∈ RN
with β independent of y and z.
Proof of Theorem 9:
endowed with the norm
Consider the product space X =
[h]0,p0 =
µX
N
i=1
for h = (h1 , .., hN ) ∈ X .
0
khi kp0,p0
¶ 10
p
QN
i=1 L
p0 (Ω)
DIRICHLET PROBLEMS WITH p-LAPLACIAN
351
We define g = (g1 , ..., gN ) : W01,p (Ω) → X by
g(u) = |∇u|p−2 ∇u ,
for u ∈ W01,p (Ω).
Let us prove that g is continuous.
By the equivalence of the norms on RN we can find a constant C1 > 0 such
that
Z
0
p0
[h]0,p0 ≤ C1 |h|p ,
Ω
for all h ∈ X .
Let p ∈ (2, ∞) and u, v ∈ W01,p (Ω).
inequality, we have:
h
g(u) − g(v)
i p0
0,p
By Lemma 1 (i) and by the Hölder
Z ¯
¯ p0
¯
¯
≤ C1 ¯g(u) − g(v)¯
Ω
≤ C2
Z
|∇u − ∇v|p
Ω
0
°
°
³
|∇u| + |∇v|
h
g(u) − g(v)
i
0
0,p
´p0 (p−2)
°p0 (p−2)
°
≤ C2 ku − vkp1,p °|∇u| + |∇v|°
which yields
(6)
0
0,p
³
≤ C ku − vkp1,p kuk1,p + kvk1,p
´p0 (p−2)
with C > 0 constant independent of u and v.
If p ∈ (1, 2] and u, v ∈ W01,p (Ω), then from Lemma 1 (ii) it follows
h
or
(7)
g(u) − g(v)
i p0
0,p0
h
≤ C20
Z
0
|∇u − ∇v|p (p−1) = C20 ku − vkp1,p
Ω
g(u) − g(v)
i
0,p0
p−1
≤ C 0 ku − vk1,p
with C 0 > 0 constant independent of u and v.
From (6) and (7) the continuity of g is obvious.
On the other hand, it holds
(8)
°
°
h
i
° 0
°
°ψ (u) − ψ 0 (v)° ≤ K g(u) − g(v)
∗
with K > 0 constant independent of u, v ∈ W01,p (Ω).
0,p0
352
G. DINCA, P. JEBELEAN and J. MAWHIN
Indeed, by the Hölder inequality and by the equivalence of the norms on RN ,
we successively have:
Z ¯
¯D
¯
E¯
¯ 0
¯
¯
¯
0
¯ ψ (u) − ψ (v), w ¯ ≤
¯g(u) − g(v)¯ |∇w|
Ω
≤
¶1
µZ ¯
¯p0 ¶ p10 µ Z
¯
¯
p p
|∇w|
g(u)
−
g(v)
¯
¯
Ω
Ω
µX
N °
°p0 ¶ p10
°
°
kwk1,p
≤ K
°gi (u) − gi (v)° 0
0,p
i=1
h
= K g(u) − g(v)
i
0,p0
kwk1,p
for u, v, w ∈ W01,p (Ω), proving (8).
Now, by the continuity of g and (8), the conclusion of the theorem follows in
a standard way: a functional is continuously Fréchet differentiable if and only if
it is continuously Gâteaux differentiable.
Remark 4. Naturally, the Fréchet differential of ψ at u ∈ W01,p (Ω) will be
denoted by ψ 0 (u) and it is clear that ψ 0 (u) = −∆p u.
Theorem 10. The operator −∆p satisfies the (S+ ) condition: if un * u
(weakly in W01,p (Ω)) and lim sup h−∆p un , un − ui ≤ 0, then un → u (strongly in
W01,p (Ω)).
n→∞
Proof: It is a simple consequence of Proposition 2, Theorems 6, 2 and 7.
2 – The problem −∆p u = f (x, u), u|∂Ω = 0
In this paragraph we are interested about sufficient conditions on the righthand member f ensuring the existence of some u ∈ W01,p (Ω) such that the equality
0
−∆p u = f (x, u) holds in the sense of W −1,p (Ω). Such an u will be called solution
of the Dirichlet problem
(9)
(
−∆p u = f (x, u)
u|∂Ω = 0 .
in Ω ,
353
DIRICHLET PROBLEMS WITH p-LAPLACIAN
0
Thus, first of all, appropriate conditions on f ensuring that Nf u ∈ W −1,p (Ω)
must be formulated, Nf being the well known Nemytskii operator defined by f ,
i.e. (Nf u)(x) = f (x, u(x)) for x ∈ Ω. Hence, we are guided to consider some basic
results on the Nemytskii operator. Simple proofs of these facts can be found in
e.g. de Figueiredo [14] or Kavian [20] (see also Vainberg [28]).
2.1. Detour on the Nemytskii operator
Let Ω be as in the beginning of Section 1.2 and f : Ω×R →R be a Carathéodory
function, i.e.:
(i) for each s ∈ R, the function x 7→ f (x, s) is Lebesgue measurable in Ω;
(ii) for a.e. x ∈ Ω, the function s 7→ f (x, s) is continuous in R.
We make the convention that in the case of a Carathéodory function, the
assertion “x ∈ Ω” to be understood in the sense “a.e. x ∈ Ω”.
Let M be the set of all measurable function u : Ω → R.
Proposition 4. If f : Ω × R →R is Carathéodory, then, for each u ∈ M,
the function Nf u : Ω → R defined by
(Nf u)(x) = f (x, u(x))
for x ∈ Ω
is measurable in Ω.
In view of this proposition, a Carathéodory function f : Ω × R →R defines an
operator Nf : M → M, which is called Nemytskii operator.
The proposition here below states sufficient conditions when a Nemytskii operator maps an Lp1 space into another Lp2 space.
Proposition 5. Suppose f : Ω × R →R is Carathéodory and the following
growth condition is satisfied:
|f (x, s)| ≤ C |s|r + b(x)
for x ∈ Ω, s ∈ R ,
where C ≥ 0 is constant, r > 0 and b ∈ Lq1 (Ω), 1 ≤ q1 < ∞.
Then Nf (Lq1 r (Ω)) ⊂ Lq1 (Ω). Moreover, Nf is continuous from Lq1 r (Ω) into
q
L 1 (Ω) and maps bounded sets into bounded sets.
354
G. DINCA, P. JEBELEAN and J. MAWHIN
Concerning the potentiality of a Nemytskii operator, we have:
Proposition 6. Suppose f : Ω × R →R is Carathéodory and it satisfies the
growth condition:
|f (x, s)| ≤ C |s|q−1 + b(x)
for x ∈ Ω, s ∈ R ,
0
where C ≥ 0 is constant, q > 1, b ∈ Lq (Ω),
1
q
+
Let F : Ω × R →R be defined by F (x, s) =
Then:
Rs
1
q0
= 1.
f (x, τ ) dτ .
0
(i) the function F is Carathéodory and there exist C1 ≥ 0 constant and
c ∈ L1 (Ω) such that
|F (x, s)| ≤ C1 |s|q + c(x)
for x ∈ Ω, s ∈ R ;
R
(ii) the functional Φ : Lq (Ω)→R defined by Φ(u) : = NF u =
Ω
R
F (x, u) is
Ω
continuously Fréchet differentiable and Φ0 (u) = NF u for all u ∈ Lq (Ω).
It should be noticed that, under the conditions of the above Proposition 6,
0
we have Nf (Lq (Ω)) ⊂ Lq (Ω), NF (Lq (Ω)) ⊂ L1 (Ω), each of the Nemytskii operators Nf and NF being continuous and bounded (it is a simple consequence of
Proposition 5). It should also be noticed that for each fixed u ∈ Lq (Ω), it holds
0
Nf u = Φ0 (u) ∈ Lq (Ω).
Now, we return to problem (9).
First, let us denote by p∗ the Sobolev conjugate exponent of p, i.e.
p∗ =

Np



if p < N ,
N −p


∞
if p ≥ N .
Below, the function f : Ω × R → R will be always assumed Carathéodory and
satisfying the growth condition
(10)
|f (x, s)| ≤ C |s|q−1 + b(x)
for x ∈ Ω, s ∈ R ,
0
where C ≥ 0 is constant, q ∈ (1, p∗ ), b ∈ Lq (Ω),
1
q
+
1
q0
= 1.
DIRICHLET PROBLEMS WITH p-LAPLACIAN
355
The restriction q ∈ (1, p∗ ) ensures that the imbedding W01,p (Ω) ,→ Lq (Ω) is
compact. Hence, the diagram
Nf
Id
Id∗
0
0
W01,p (Ω) ,→ Lq (Ω) → Lq (Ω) ,→ W −1,p (Ω)
shows that Nf is a compact operator (continuous and maps bounded sets into
0
relatively compact sets) from W01,p (Ω) into W −1,p (Ω).
An element u ∈ W01,p (Ω) is said to be solution of problem (9) if
− ∆p u = N f u
(11)
0
in the sense of W −1,p (Ω) i.e.
h−∆p u, vi = hNf u, vi
or
Z
(12)
|∇u|
Ω
p−2
∇u ∇v =
Z
for all v ∈ W01,p (Ω)
f (x, u) v
for all v ∈ W01,p (Ω) .
Ω
At this stage, in the approach of problem (9), two strategies appear to be
natural.
The first reduces problem (9) to a fixed point problem with compact operator.
0
Indeed, by Theorem 8, the operator (−∆p )−1 : W −1,p (Ω) → W01,p (Ω) is
bounded and continuous.
Consequently, (11) can be equivalently written
u = (−∆p )−1 Nf u
(13)
with (−∆p )−1 Nf : W01,p (Ω) → W01,p (Ω) a compact operator.
The second is a variational one: the solutions of problem (9) appear as critical
points of a C 1 functional on W01,p (Ω).
To see this, we first have that −∆p = ψ 0 , where the functional ψ(u) = p1 kukp1,p
is continuously Fréchet differentiable on W01,p (Ω). On the other hand, under the
basic condition (10) and taking into account that the imbedding W01,p (Ω) →
Lq (Ω) is continuous (in fact, compact), the functional Φ : W01,p (Ω) → R defined
by Φ(u) =
tiable on
R
F (x, u) with F (x, s) =
Ω
W01,p (Ω)
Rs
0
and Φ0 (u) = Nf u.
f (x, τ ) dτ , is continuously Fréchet differen-
356
G. DINCA, P. JEBELEAN and J. MAWHIN
Consequently, the functional F : W01,p (Ω) → R defined by
1
F(u) = ψ(u) − Φ(u) = kukp1,p −
p
Z
F (x, u)
Ω
is C 1 in W01,p (Ω) and
F 0 (u) = (−∆p )u − Nf u .
The search for solutions of problem (9) is, now, reduced to the search of critical
points of F, i.e. of those u ∈ W01,p (Ω) such that F 0 (u) = 0.
2.2. Existence of fixed points for (−∆p )−1 Nf via a Leray–Schauder
technique
In this section, the “a priori estimate method” will be used in order to establish the existence of fixed points for the compact operator T = (−∆p )−1 Nf :
W01,p (Ω) → W01,p (Ω) (see Dinca and Jebelean [13]).
For it suffices to prove that the set
S =
n
u ∈ W01,p (Ω) | u = α T u for some α ∈ [0, 1]
o
is bounded in W01,p (Ω).
By (10), for arbitrary u ∈ W01,p (Ω), it is obvious that
kT ukp1,p =
≤
D
(−∆p )T u, T u
Z ³
Ω
C |u|
q−1
E
= hNf u, T ui =
´
Z
f (x, u) T u
Ω
+ b(x) |T u| .
Furthermore, for u ∈ S i.e. u = α T u, with some α ∈ [0, 1], we have
kT ukp1,p ≤ C αq−1 kT ukq0,q + kbk0,q0 kT uk0,q
≤ C αq−1 C1q kT ukq1,p + kbk0,q0 C1 kT uk1,p
≤ C C1q kT ukq1,p + kbk0,q0 C1 kT uk1,p
the constant C1 coming from the continuous imbedding W01,p (Ω) → Lq (Ω).
DIRICHLET PROBLEMS WITH p-LAPLACIAN
357
Consequently, for each u ∈ S, it holds
(14)
kT ukp1,p − K1 kT ukq1,p − K2 kT uk1,p ≤ 0
with K1 , K2 ≥ 0 constants.
Remark that if (14) would imply that there is a constant a ≥ 0 such that
kT uk1,p ≤ a, then the boundedness of S would be proved, because we would have
kuk1,p = α kT uk1,p ≤ a.
But this is obviously true if q ∈ (1, p).
We have obtained
Theorem 11. If the Carathéodory function f : Ω × R → R satisfies (10)
with q ∈ (1, p) then the operator (−∆p )−1 Nf has fixed points in W01,p (Ω) or
equivalently, problem (9) has solutions. Moreover, the set of all solutions of
problem (9) is bounded in the space W01,p (Ω).
Remark 5. We shall see that if (10) holds with b ∈ L∞ (Ω) then the variational approach allows to weaken the hypotheses of Theorem 11 and problem
(9) still has solutions but the boundedness of the set of all solutions will not be
ensured.
Remark 6. The condition q ∈ (1, p) appear as a technical condition, needed
in obtaining the boundedness of S.
It is a natural question if the set S still remains bounded in case that q = p
and it is a simple matter to see that if q = p and 1 − K1 > 0 then the above
reasoning still works. This means that we are interested to work with “the best
constants” C and C1 such that 1 − C · C1p be strictly positive.
There are situations when 1 − C · C1p > 0 fails. The example here below shows
that then S can be unbounded.
Let λ be an eigenvalue of −∆p in W01,p (Ω) and u be a corresponding eigenvector:
−∆p u = λ |u|p−2 u .
It is clear that
(15)
kukp1,p
.
λ =
kukp0,p
Because kvk0,p ≤ C1 kvk1,p for all v ∈ W01,p (Ω), from (15), it results that 1−λC1p ≤ 0.
358
G. DINCA, P. JEBELEAN and J. MAWHIN
Consider the Carathéodory function f (x, s) = λ |s|p−2 s. Clearly, the growth
condition (10) is satisfied with q = p and b = 0, c = λ. Consequently, (14)
becomes
(1 − λ C1p ) kT ukp1,p ≤ 0
for all u ∈ S and no conclusion on the boundedness of S can be derived.
In fact, S is unbounded.
Indeed, we have −∆p (t u) = Nf (t u) i.e. t u = (−∆p )−1 Nf (t u) for all t ∈ R,
which means {t u | t ∈ R} ⊂ S and so, S is unbounded.
Remark 7. In the case f (x, s) = g(s) + h(x) with g : R → R continuous
and h ∈ L∞ (Ω), the homotopy invariance of Leray–Schauder degree (but in a
different functional framework) is used by Hachimi and Gossez [17] in order to
prove the following result (see [17] Th 1.1):
If
g(s)
pG(s)
(i) lim sup p−2 ≤ λ1 and (ii) lim sup
< λ1
s
|s|p
s→±∞ |s|
s→±∞
where G(s) =
the problem
Rs
0
g(τ ) dτ and λ1 is the first eigenvalue of −∆p in W01,p (Ω), then
−∆p u = g(u) + h(x)
in Ω, u = 0 on ∂Ω
has a solution in W01,p (Ω) ∩ L∞ (Ω).
2.3. Existence results by a direct variational method
As we have already emphasized in Section 2.1, in the variational approach,
under the growth condition (10) on f , the solutions of problem (9) are precisely
the critical points of the C 1 functional F : W01,p (Ω) → R defined by
1
F(u) = ψ(u) − Φ(u) = kukp1,p −
p
where F (x, u) =
Rs
0
Z
F (x, u)
Ω
f (x, τ ) dτ .
Remark that the compact imbedding W01,p (Ω) ,→ Lq (Ω) implies that F is
weakly lower semicontinuous in W01,p (Ω) .
So, by a standard result, in order to derive sufficient conditions for (9) has
solutions, a first suitable way is to ensure the coerciveness of F. Such results
DIRICHLET PROBLEMS WITH p-LAPLACIAN
359
were obtained by Anane and Gossez [4] even in more general conditions on f . It
is not our aim to detail this direction. However, we depict a few such of results.
First we refer to a result of Anane and Gossez [4].
Let G : Ω × R → R be a Carathéodory function, such that, for any R > 0,
Ω 3 x → sup |G(x, s)| ∈ L1 (Ω) .
(16)
|s|≤R
λ1 |s|p
p
We write G(x, s) =
W01,p (Ω)
+ H(x, s), where λ1 is the first eigenvalue of −∆p
on
(see e.g. Anane [3], Lindqvist [22]) and let us define H ± (x) as the
superior limit of H(x,s)
|s|p as s → ±∞ respectively.
It holds (see Proposition 2.1 in Anane–Gossez [4]):
Theorem 12. Assume (16) and
(i) H ± (x) ≤ 0 a.e. uniformly in x;
(ii) H + (x) < 0 on Ω+ and H − (x) < 0 on Ω− for subsets Ω± of positive
measure.
Then
G(u) =
1
kukp1,p −
p
Z
G(x, u)
Ω
is well defined on W01,p (Ω), takes values in ]− ∞, +∞], is weakly lower semicontinuous and coercive.
We now return to problem (9).
By virtue of Proposition 6 (i), we have for any R > 0
sup |F (x, s)| ≤ C1 Rq + c(x) ∈ L1 (Ω)
|s|≤R
showing that (16) is fulfilled with G(x, s) = F (x, s).
p
Clearly, in this case H(x, s) = F (x, s) − λ1p|s| .
In order to extend a result of Mawhin–Ward–Willem [25] for the particular
case p = 2 to the general case p ∈ (1, ∞), suppose that there exists a function
α(x) ∈ L∞ (Ω) with α(x) < λ1 , on a set of positive measure, such that
(17)
lim sup
s→±∞
p F (x, s)
≤ α(x) ≤ λ1
|s|p
uniformly in Ω .
360
G. DINCA, P. JEBELEAN and J. MAWHIN
We obtain
µ
F (x, s) λ1
H(x, s)
= lim sup
−
p
|s|
|s|p
p
s→±∞
s→±∞
F (x, s) λ1
α(x) − λ1
= lim sup
≤
−
p
|s|
p
p
s→±∞
H ± (x) = lim sup
¶
=
which yields H ± (x) ≤ 0 uniformly in Ω, i.e. (i) in Theorem 12.
On the other hand, it is clear that
H ± (x) ≤
α(x) − λ1
< 0
p
on the set of positive measure Ω1 = {x ∈ Ω | α(x) < λ1 } and (ii) in Theorem 12
is checked.
We have obtained
Theorem 13. Let f : Ω × R → R be a Carathéodory function satisfying the
growth condition (10). Suppose that there exists α(x) ∈ L∞ (Ω) with α(x) < λ1
on a set of positive measure such that (17) holds.
Then F is coercive; consequently problem (9) has solutions.
A direct proof of Theorem 13 can be given as it follows.
Define N : W01,p (Ω) → R by
N (v) = kvkp1,p −
Z
α(x) |v|p
Ω
and let us prove that there exists ε0 > 0 such that
(18)
N (v) ≥ ε0
for all v ∈ W01,p (Ω) with kvk1,p = 1 .
For, let us recall (see e.g. Anane [3]) that
(19)
λ1 = inf
(
kvkp1,p
| v ∈ W01,p (Ω)\{0}
kvkp0,p
)
the infimum being attained exactly when v is multiple of some function u1 > 0.
By (17) and (19) it follows that N (v) ≥ 0 for all v ∈ W01,p (Ω).
Supposing, by contradiction, that there is a sequence (vn ) in W01,p (Ω) with
kvn k1,p = 1 and N (vn ) → 0, we can find a subsequence of (vn ), still denoted by
(vn ), and some v0 ∈ W01,p (Ω) with vn * v0 , weakly in W01,p (Ω) and vn → v0 ,
strongly in Lp (Ω).
DIRICHLET PROBLEMS WITH p-LAPLACIAN
The functional v 7→
R
361
α(x) |v|p is continuous on Lp (Ω) and weakly continuous
Ω
on W01,p (Ω).
By the weakly lower semicontinuity of N on W01,p (Ω), we infer
0 ≤
kv0 kp1,p
Z
− α(x) |v0 |p ≤ lim inf N (vn ) = 0
n→∞
Ω
and so, kv0 kp1,p =
R
R
α(x) |v0 |p . But N (vn ) → 1 − α(x) |v0 |p , hence
Ω
Ω
kv0 kp1,p =
Z
α(x) |v0 |p = 1
Ω
which yields v0 6= 0.
Then, by (17) and (19) we have
λ1 kv0 kp0,p ≤ kv0 kp1,p =
(20)
Z
α(x) |v0 |p ≤ λ1 kv0 kp0,p
Ω
which implies that λ1 =
kv0 kp1,p
.
kv0 kp0,p
It results that v0 is a nonzero multiple of u1 .
Consequently, |v0 (x)| > 0 a.e. in Ω.
But, then, denoting Ω1 : = {x ∈ Ω | α(x) < λ1 }, because meas(Ω1 ) > 0, we get
Z
α(x) |v0 |p =
Ω
Z
α(x) |v0 |p +
Ω1
Z
α(x) |v0 |p < λ1 kv0 kp0,p
Ω\Ω1
contradicting (20). So, (18) is proved.
Obviously, from (18) we have
(21)
Z
kvkp1,p − α(x) |v0 |p ≥ ε0 kvkp1,p
for all v ∈ W01,p (Ω) .
Ω
Let ε > 0 be such that ε < λ1 ε0 .
Using (17) and by Proposition 6 (i) a straightforward computation shows that
there exists a constant k = k(ε) such that
(22)
F (x, s) ≤
α(x) + ε p
|s| + k + c(x)
p
for x ∈ Ω, s ∈ R .
362
G. DINCA, P. JEBELEAN and J. MAWHIN
Now, by (21) and (22) we estimate F as it follows
´
1³
ε0 kvkp1,p − ε kvkp0,p − k1
p
λ 1 ε0 − ε
kvkp1,p − k1 → ∞
≥
p
F(v) ≥
as kvk1,p → ∞.
Remark 8.
(i) If in (10) the function b is required to be in L∞ (Ω) then it is easy to
check that if q ∈ (1, p) then (17) holds with α ≡ 0 and so, problem
(9) has solutions. But, as we have already remarked (see Remark 5), if
b ∈ L∞ (Ω) and only (17) is required then the boundedness of the set of
solutions (as in Theorem 11) is not stated.
(ii) The idea of the above direct proof of Theorem 13 is a suitable one in
proving the existence of solutions for a multivalued variant of problem
(9) (see Proposition 4.1 in Jebelean [19]).
2.4. Using the Mountain Pass Theorem
(cf. Dinca, Jebelean and Mawhin [12])
Again, the Carathéodory function f : Ω × R → R is assumed to satisfy the
growth condition (10).
The existence of nontrivial critical points of the C1 functional F : W01,p (Ω)→R
means that the Dirichlet problem (9) has nontrivial solutions.
This section is devoted to formulate supplementary conditions on f and F
ensuring the existence of such nontrivial critical points for F.
The main tool in this direction is the well known “Mountain Pass Theorem”
of Ambrosetti and Rabinowitz [2] which we recall here in a useful and popular
form (see e.g. Theorem 2.2 in Rabinowitz [26]).
Theorem 14. Let X be a real Banach space and I ∈ C1 (X, R) satisfying
the Palais–Smale (PS) condition. Suppose I(0) = 0 and
(I1 ) there are constants ρ, α > 0 such that I|kxk=ρ ≥ α;
(I2 ) there is an element e ∈ X, kek > ρ such that I(e) ≤ 0.
363
DIRICHLET PROBLEMS WITH p-LAPLACIAN
Then I possesses a critical value c ≥ α. Moreover c can be characterized as
c = inf
max I(u)
g∈Γ u∈g([0,1])
where
Γ =
n
o
g ∈ C([0, 1], X) | g(0) = 0, g(1) = e .
It is obvious that each critical point u at level c (I 0 (u) = 0, I(u) = c) is a
nontrivial one. Consequently, if the hypothesis of Theorem 14 are satisfied with
X = W01,p (Ω) and I = F then the existence of nontrivial solutions for problem
(9) is ensured.
We first deal with the (PS) condition for F.
Recall that F is said to satisfy the (PS) condition if any sequence (un ) ⊂
1,p
W0 (Ω) for which F(un ) is bounded and F 0 (un ) → 0 as n → ∞, possesses a
convergent subsequence.
Lemma 2. If (un ) ⊂ W01,p (Ω) is bounded and F 0 (un ) → 0 as n → ∞, then
(un ) has a convergent subsequence.
Proof: One can extract a subsequence (unk ) of (un ), weakly convergent to
some u ∈ W01,p (Ω). As F 0 (unk ) → 0, we infer
(23)
But
D
F 0 (unk ), unk − u
E
=
D
−∆p unk − Nf unk , unk − u
E
→ 0.
hNf unk , unk − ui → 0
because of
¯
¯
¯
¯
¯hNf unk , unk − ui¯ ≤ kNf unk ko,q 0 kunk − uk0,q
and, by unk * u in W01,p (Ω) and by the compact imbedding W01,p (Ω) ,→ Lq (Ω),
0
we get unk → u strongly in Lq (Ω). Notice that (Nf unk ) is bounded in Lq (Ω).
By (23) we obtain
h−∆p unk , unk − ui → 0
which, together with Theorem 10 shows that unk → u strongly in W01,p (Ω).
Theorem 15. If there exist θ > p and s0 > 0 such that
(24)
θ F (x, s) ≤ s f (x, s)
then F satisfies the (PS) condition.
for x ∈ Ω, |s| ≥ s0 ,
364
G. DINCA, P. JEBELEAN and J. MAWHIN
Remark 9. It is worth noticing that (24) extends the well known condition
there are θ > 2 and s0 > 0 such that
0 < θ F (x, s) ≤ s f (x, s)
for x ∈ Ω, |s| ≥ s0
which was first formulated by Ambrosetti and Rabinowitz [2] as a sufficient condition to ensure that F satisfies (PS) in the particular case p = 2.
Proof of Theorem 15: It suffices to show that any sequence (un ) ⊂ W01,p (Ω)
for which (F(un )) is bounded and F 0 (un ) → 0, is bounded. Then Lemma 2 will
accomplish the proof.
Let d ∈ R be such that F(un ) ≤ d for all n ∈ N. For each n ∈ N, we denote
n
o
Ωn = x ∈ Ω | |un (x)| ≥ s0 ,
We have
(25)
1
kun kp1,p −
p
µ Z
F (x, un ) +
Z
Ω0n = Ω\Ωn .
F (x, un )
Ω0n
Ωn
¶
≤ d.
We proceed with obtaining estimations independent of n for the integrals in
(25).
Let n ∈ N be arbitrary chosen.
If x ∈ Ω0n , then |un (x)| < s0 and by Proposition 6 (i), it follows
F (x, un ) ≤ C1 |un (x)|q + c(x) ≤ C1 sq0 + c(x)
and hence
(26)
Z
F (x, un ) ≤
C1 sq0
Z
· meas(Ω) + c(x) = K1 .
Ω0n
Ω
If x ∈ Ωn , then |un (x)| ≥ s0 and by (24) it holds
F (x, un ) ≤
1
f (x, un (x)) un (x)
θ
which gives
(27)
Z
F (x, un ) ≤
Ωn
Z
Ωn
1
1
f (x, un ) un =
θ
θ
µZ
f (x, un ) un −
Z
f (x, un ) un
Ω0n
Ω
By the growth condition (10), we deduce
¯Z
¯
Z ³
´
¯
¯
q
¯ f (x, un ) un ¯ ≤
C
|u
|
+
b(x)
|u
|
n
n
¯
¯
Ω0n
Ω0n
≤ C sq0 · meas(Ω) + s0
Z
Ω
b(x) = K2
¶
.
DIRICHLET PROBLEMS WITH p-LAPLACIAN
365
which yields
−
(28)
1
θ
Z
f (x, un ) un ≤
Ω0n
K2
.
θ
Finally, by (25), (26), (27) and (28) we get
1
1
kun kp1,p −
p
θ
Z
f (x, un ) un ≤ d + K1 +
Ω
K2
= K ,
θ
1
1
kun kp1,p − hNf un , un i ≤ K .
p
θ
(29)
On the other hand, because F 0 (un ) → 0 as n → ∞, there is n0 ∈ N such that
|hF 0 (un ), un i| ≤ kun k1.p for n ≥ n0 . Consequently, for all n ≥ n0 , we have
¯
¯
¯
¯
¯h−∆p un , un i − hNf un , un i¯ ≤ kun k1,p
or
¯
¯
¯
¯
p
¯kun k1.p − hNf un , un i¯ ≤ kun k1,p
which gives
1
1
1
kun kp1,p − kun k1,p ≤ − hNf un , un i .
θ
θ
θ
Now, from (29) and (30) it results
−
(30)
µ
¶
1
1 1
kun kp1,p − kun k1,p ≤ K
−
p θ
θ
and taking into account that θ > p, we conclude that (un ) is bounded.
Now, viewing (I2 ) in Theorem 14, the next step is to obtain sufficient conditions for F be unbounded below in W01,p (Ω). The following lemma will throw
light in the role of this unboundedness.
Lemma 3. The functional F has the properties:
(i) F(0) = 0;
(ii) F maps bounded sets into bounded sets.
Proof: (i) Obvious.
(ii) Because
F 0 (u) = −∆p u − Nf u
366
G. DINCA, P. JEBELEAN and J. MAWHIN
it is clear that
kF 0 (u)k∗ ≤ k − ∆p uk∗ + kNf uk∗
p−1
≤ kuk1,p
+ K kNf uk0,q0
for all u ∈ W01,p (Ω).
Furthermore, by the (compact) imbedding W01,p (Ω) ,→ Lq (Ω) and by virtue
0
of the fact that Nf maps bounded sets in Lq (Ω) into bounded sets in Lq (Ω), we
0
conclude that F 0 maps bounded sets in W01,p (Ω) into bounded sets in W −1,p (Ω).
Let v be arbitrary chosen in W01,p (Ω). We have:
|F(v)| = |F(v) − F(0)| = |hF 0 (ξv), vi| ≤ kF 0 (ξv)k∗ kvk1,p
with ξ ∈ (0, 1). Then (ii) follows by the above conclusion on F 0 .
Remark 10. Actually, (ii) in the above Lemma 3 is a consequence of the
p−1
growth condition (10) and of the fact that k− ∆p uk∗ = kuk1,p
.
Remark 11. Assume F is unbounded from below. Then, for any ρ > 0
there is an element e ∈ W01,p (Ω) with kek1,p ≥ ρ, such that F(e) ≤ 0.
Indeed, suppose by contradiction that there is some ρ > 0 such that for all
u ∈ W01,p (Ω) with kuk ≥ ρ, it holds F(u) ≥ 0. Then by Lemma 3 (ii), the set
{F(u) | kuk1,p < ρ} is bounded. It results that F is bounded from below, which
is a contradiction.
Theorem 16. If either
(i) there are numbers θ > p and s1 > 0 such that
(31)
0 < θ F (x, s) ≤ s f (x, s)
for x ∈ Ω, s ≥ s1 ,
or
(ii) there are numbers θ > p and s1 < 0 such that
(32)
0 < θ F (x, s) ≤ s f (x, s)
for x ∈ Ω, s ≤ s1 ,
then F is unbounded from below.
Proof: We shall prove the sufficiency of condition (i) (similar argument if
(ii) holds).
367
DIRICHLET PROBLEMS WITH p-LAPLACIAN
More precisely, we’ll show that if u ∈ W01,p (Ω), u > 0 is such that
meas(M1 (u)) > 0 holds, with
M1 (u) =
then F(λu) → −∞ as λ → ∞.
First, for λ ≥ 1, let us denote
Mλ (u) =
o
n
x ∈ Ω | u(x) ≥ s1 ,
n
o
x ∈ Ω | λ u(x) ≥ s1
and let us remark that M1 (u) ⊂ Mλ (u), and hence meas(Mλ (u)) > 0.
On the other hand, there is a function γ ∈ L1 (Ω), γ > 0 such that
F (x, s) ≥ γ(x) sθ
(33)
for x ∈ Ω, s ≥ s1 .
Indeed, for x ∈ Ω and τ ≥ s1 , by (31) we have
f (x, τ )
F 0 (x, τ )
θ
≤
= τ
τ
F (x, τ )
F (x, τ )
and integrating from s1 to s we get
ln
µ
s
s1
¶θ
≤ ln F (x, s) − ln F (x, s1 )
F (x,s1 )
sθ1
which implies (33) with γ(x) =
Now, let λ ≥ 1. Clearly,
(34)
F(λu) =
λp
kukp1,p −
p
µ Z
> 0.
F (x, λu) +
Mλ (u)
Z
F (x, λu)
Ω\Mλ (u)
¶
.
If x ∈ Mλ (u) then λu(x) ≥ s1 , and by (33)
F (x, λu(x)) ≥ γ(x) λθ uθ .
Therefore,
(35)
Z
F (x, λu) ≥ λθ
Mλ (u)
Z
γ(x) uθ ≥ λθ
Mλ (u)
Z
γ(x) uθ = λθ K1 (u) ,
M1 (u)
with K1 (u) > 0.
If x ∈ Ω\Mλ (u) then λu(x) < s1 , and by virtue of Proposition 6 (i), we
obtain
|F (x, λu(x))| ≤ C1 λq uq + c(x) ≤ C1 sq1 + c(x) .
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G. DINCA, P. JEBELEAN and J. MAWHIN
Therefore,
¯
¯
¯
¯
(36)
¯
Z
¯
F (x, λu) ¯¯ ≤ C1 sq1 · meas(Ω) + c(x) = K2 .
Z
Ω
Ω\Mλ (u)
From (34), (35) and (36) it results
F(λu) ≤
λp
kukp1,p − λθ K1 (u) + K2 → −∞
p
as λ → ∞
and the proof is complete.
Concerning condition (I1 ) in Theorem 14, we have the following
Theorem 17. Suppose the Carathéodory function f : Ω × R → R satisfies
(i) there is q ∈ (1, p∗ ) such that
|f (x, s)| ≤ C(|s|q−1 + 1)
for x ∈ Ω, s ∈ R ,
with C ≥ 0 constant;
(ii)
lim sup
s→0
f (x, s)
< λ1
|s|p−2 s
uniformly with x ∈ Ω
where λ1 is the first eigenvalue of −∆p on W01,p (Ω).
Then there are constants ρ, α > 0 such that F|kuk1,p =ρ ≥ α.
Proof: We define h : Ω → R by putting
h(x) = lim sup
s→0
f (x, s)
.
|s|p−2 s
By (ii) we can find µ ∈ (0, λ1 ) such that h(x) < µ uniformly with x ∈ Ω.
Therefore, there is some δµ > 0 such that
f (x, s)
≤ µ
|s|p−2 s
for x ∈ Ω, 0 < |s| < δµ ,
or
(37)
f (x, s) ≤ µ sp−1
(38)
− µ |s|p−1 ≤ f (x, s)
for x ∈ Ω, s ∈ (0, δµ ) ,
for x ∈ Ω, s ∈ (−δµ , 0) .
Remark that the Carathéodory function f satisfies f (x, 0) = 0 for x ∈ Ω.
DIRICHLET PROBLEMS WITH p-LAPLACIAN
369
From (37), (38) and by the definition of F , we infer
F (x, s) ≤
(39)
µ p
|s|
p
for x ∈ Ω, |s| < δµ .
Taking into account (i), it is easy to see that F satisfies
|F (x, s)| ≤ C1 (|s|q + 1)
(40)
for x ∈ Ω, s ∈ R
with C1 ≥ 0 constant.
Choose q1 ∈ (max{p, q}, p∗ ). Then by (40), there is a constant C2 ≥ 0 such
that
(41)
|F (x, s)| ≤ C2 |s|q1 for x ∈ Ω, |s| ≥ δµ .
From (39) and (41), we have
F (x, s) ≤
(42)
µ p
|s| + C2 |s|q1
p
for x ∈ Ω, s ∈ R .
Now, by the variational characterization of the first eigenvalue λ1 (see (19)),
by the estimate (42) and by the imbedding W01,p (Ω) ,→ Lq1 (Ω), it results
F(u) =
1
kukp1,p −
p
Z
Ω
1
µ
≥
kukp1,p −
p
p
F (x, u)
Z
p
|u| − C2
Ω
Z
|u|q1
Ω
µ
1
1
kukp1,p − kukp0,p − C3 kukq1,p
≥
p
p
#
" µ
¶
kukp0,p
1
q1 −p
p
1−µ
− C3 kuk1,p
= kuk1,p
p
kukp1,p
≥
kukp1,p
"
µ
1
µ
1−
p
λ1
¶
−
q1 −p
C3 kuk1,p
#
≥ α > 0,
provided kuk1,p = ρ is sufficiently small.
The following lemma will be needed in the sequel.
Lemma 4.
(i) If u ∈ W01,p (Ω) is a solution of problem (9) with f (x, s) ≥ 0 for x ∈ Ω
and s ≤ 0, then u ≥ 0.
(ii) If u ∈ W01,p (Ω) is a solution of problem (9) with f (x, s) ≤ 0 for x ∈ Ω
and s ≥ 0, then u ≤ 0.
370
G. DINCA, P. JEBELEAN and J. MAWHIN
Proof: We shall prove (i) (similar argument for (ii)).
Let u ∈ W01,p (Ω) be a solution of problem (9) and let us denote Ω− = {x ∈ Ω |
u(x) < 0}. We define u− = max{−u, 0}. By Theorem A.1 in Kinderlehrer and
Stampacchia [21], it is known that u− ∈ W01,p (Ω) and it is obvious that
∇u− =
From
(
−∇u
in Ω− ,
0
in Ω\Ω− .
Z
|∇u|p−2 ∇u ∇u− =
−
Z
Ω
Z
f (x, u) u−
Ω
we obtain
p
|∇u| = −
Ω−
Z
f (x, u) u ≥ 0 .
Ω−
Thus, ∇u = 0 a.e. in Ω− , consequently ∇u− = 0 a.e. in Ω.
ku− k1,p = 0 or u− = 0 a.e. in Ω.
We conclude that meas(Ω− ) = 0, i.e. u ≥ 0 a.e. in Ω.
Therefore,
At this stage we are in position to prove the main result of this section.
Theorem 18. Suppose f : Ω × R →R is Carathéodory and it satisfies:
(i) there is q ∈ (1, p∗ ) such that
|f (x, s)| ≤ C(|s|q−1 + 1)
for x ∈ Ω, s ∈ R ,
with C ≥ 0 constant;
(ii)
lim sup
s→0
f (x, s)
< λ1
|s|p−2 s
uniformly with x ∈ Ω
where λ1 is the first eigenvalue of −∆p on W01,p (Ω);
(iii) there are constants θ > p and s0 > 0 such that
0 < θ F (x, s) ≤ s f (x, s)
for x ∈ Ω, |s| ≥ s0 .
Then problem (9) has nontrivial solutions u− ≤ 0 ≤ u+ .
Proof: We shall prove that (9) has a nontrivial solution u+ ≥ 0 (similar
argument for the existence of u− ).
DIRICHLET PROBLEMS WITH p-LAPLACIAN
371
We define f+ : Ω × R → R by f+ (x, s) = f (x, s+|s|
2 ) i.e.
f+ (x, s) =
(
if s ≤ 0 ,
if s > 0 ,
0
f (x, s)
and let F+ : Ω × R → R be defined by
F+ (x, s) =
Zs
f+ (x, τ ) dτ .
0
The following assertions are true:
(i)+ the function f+ is Carathéodory and it satisfies
|f+ (x, s)| ≤ C(|s|q−1 + 1)
(ii)+ lim sup
s→0
f+ (x,s)
|s|p−2 s
for x ∈ Ω, s ∈ R ;
< λ1 uniformly with x ∈ Ω;
(iii)+ θ F+ (x, s) ≤ s f+ (x, s) for x ∈ Ω, |s| ≥ s0 ;
(iv)+ 0 < θ F+ (x, s) ≤ s f+ (x, s) for x ∈ Ω, s ≥ s0 .
Indeed, (i)+ , (iii)+ and (iv)+ are easily seen.
To see (ii)+ , we have
(
f+ (x, s)
f+ (x, s)
f+ (x, s)
lim sup
= max lim sup
, lim sup
p−2
p−2
|s|
s
|s|
s
|s|p−2 s
s→0
s%0
s&0
(
f (x, s)
= max 0, lim sup p−2
|s|
s
s&0
)
< λ1
)
uniformly with x ∈ Ω .
From (i)+ –(iv)+ we infer that the C1 functional F+ : W01,p (Ω)→ R defined by
Z
1
F+ (u) = kukp1,p − F+ (x, u)
p
Ω
has a nontrivial critical point u+ ∈ W01,p (Ω).
To see this, we apply Theorem 14 with I = F+ .
To this end, first of all, it will be remarked that viewing (i)+ , the results
concerning F remain valid for F+ , with f+ instead of f .
Clearly, F+ (0) = 0.
By (i)+ , (ii)+ and Theorem 17, there are constants α, ρ > 0 such that
F+ |kuk1,p =ρ ≥ α.
372
G. DINCA, P. JEBELEAN and J. MAWHIN
Furthermore, by (iv)+ , Theorem 16 (i) and Lemma 3 (ii) (also see Remark 11),
there is an element e ∈ W01,p (Ω) with kek1,p ≥ ρ, such that F+ (e) ≤ 0.
Finally, by (iii)+ and Theorem 15, F+ satisfies the (PS) condition.
The nontrivial critical point u+ ∈ W01,p (Ω), whose existence is ensured by
Theorem 14, satisfies
(43)
Z
|∇u+ |p−2 ∇u+ ∇v =
Ω
Z
f+ (x, u+ ) v
for all v ∈ W01,p (Ω) .
Ω
As f+ (x, s) = 0 for x ∈ Ω, s ≤ 0, Lemma 4 (i) shows that u+ ≥ 0.
Now, by the definition of f+ , (43) becomes
Z
|∇u+ |
p−2
∇u+ ∇v =
Ω
Z
f (x, u+ ) v
for all v ∈ W01,p (Ω)
Ω
and the proof is complete.
Remark 12. Theorem 18 was originally stated by Ambrosetti and Rabinowitz (see Corollary 3.11 in Ambrosetti–Rabinowitz [2]), in the case p = 2.
After, their result became frequently cited as a typical existence result for nonlinear Dirichlet problems with right-hand member having a superlinear growth
(see e.g. Corollary 2.23 in Rabinowitz [26], Theorem 6.9 in de Figueiredo [14],
Theorem 6.2 in Struwe [27], et. al.).
In this context, Theorem 18 can be seen as a model of existence result for
Dirichlet problems with p-Laplacian having the right-hand member a function
with “super p − 1 polynomial” growth, condition (iii) implying
(44)
f (x, s)
= +∞ .
|s|→∞ |s|p−2 s
lim
Moreover, (44) shows that the generality of Theorem 18 is not lost if in (i) q is
required to be in (p, p∗ ) instead of (1, p∗ ).
On the other hand, a reasoning similar to that in the proof of Theorem 16
shows that conditions (iii) and (i) in Theorem 18 yield the existence of some
γ ∈ L∞ (Ω), γ > 0, such that F (x, s) ≥ γ(x) |s|θ for x ∈ Ω, and |s| ≥ s0 (also see
the proof of Proposition 7 bellow). This shows that the potential F grows faster
than |s|p with |s| → ∞. For an existence result allowing F to grow faster than
|s|p or slower than |s|p we refer the reader to Costa and Magalhaes [10].
DIRICHLET PROBLEMS WITH p-LAPLACIAN
373
2.5. Multiple solutions
Taking into account the minimax methods in critical point theory, invoking
the “Mountain Pass Theorem” in order to prove existence of nontrivial solutions
for problem (9), make natural the question: what about multiple solutions?
More precisely, following the particular case p = 2, it would be expected
that under the basic hypothesis of Theorem 18, the oddness of f be sufficient to
guarantee the existence of an unbounded sequence of solutions for problem (9).
Such a result will conclude the paper.
We need the following
Proposition 7. Suppose the Carathéodory function f : Ω× R → R satisfies
(i) there is q ∈ (1, p∗ ) such that
|f (x, s)| ≤ C(|s|q−1 + 1)
for x ∈ Ω, s ∈ R ,
with C ≥ 0 constant;
(ii) there are numbers θ > ρ and s0 > 0 such that
0 < θ F (x, s) ≤ s f (x, s)
for x ∈ Ω, |s| ≥ s0 .
Then, if X1 is a finite dimensional subspace of W01,p (Ω), the set
S = {v ∈ X1 | F(v) ≥ 0} is bounded in W01,p (Ω).
Proof: From (i), F satisfies
(45)
|F (x, s)| ≤ C1 (|s|q + 1)
for x ∈ Ω, s ∈ R ,
with C1 ≥ 0 constant.
We claim that there is γ ∈ L∞ (Ω), γ > 0 on Ω, such that
(46)
F (x, s) ≥ γ(x) |s|θ
for x ∈ Ω, |s| ≥ s0 .
Indeed, as in the proof of Theorem 16, we have
(47)
F (x, s) ≥ γ1 (x) sθ
for x ∈ Ω, s ≥ s0 ,
0)
where γ1 (x) = F (x,s
. By virtue of (45), it is obvious that γ1 ∈ L∞ (Ω) and (ii)
sθ0
yields γ1 > 0 on Ω.
A similar reasoning shows that
(48)
where γ2 (x) =
F (x, s) ≥ γ2 (x) |s|θ
F (x,−s0 )
.
sθ0
for x ∈ Ω, s ≤ −s0 ,
Again γ2 ∈ L∞ (Ω) and γ2 > 0 on Ω.
374
G. DINCA, P. JEBELEAN and J. MAWHIN
Therefore, (46) holds with γ(x) = min{γ1 (x), γ2 (x)} for x ∈ Ω, as claimed.
We shall prove that F satisfies
Z
1
F(v) ≤ kvkp1,p − γ(x) |v|θ − K for all v ∈ W01,p (Ω)
(49)
p
Ω
with K ≥ 0 constant.
Let v be arbitrary chosen in W01,p (Ω) and let us denote Ω< = {x ∈ Ω |
|v(x)| < s0 }.
By (45) we have
Z
F (x, v) ≥ −C1
Z
(|v|q + 1) ≥ −C1
=
−C1 (sq0
and by (46) it holds
Z
+ 1) · meas(Ω) = K1
γ(x) |v|θ .
Ω\Ω<
Then
≤
Z
F (x, v) ≥
Ω\Ω<
F(v) =
(sq0 + 1) =
Ω
Ω<
Ω<
Z
1
kvkp1,p −
p
µZ
1
kvkp1,p −
p
Z
F (x, v) +
Ω<
1
kvkp1,p −
≤
p
Z
¶
γ(x) |v|θ − K1
γ(x) |v|θ +
Ω
Z
F (x, v)
Ω\Ω<
Ω\Ω<
1
=
kvkp1,p −
p
Z
Z
γ(x) |v|θ − K1
Ω<
γ(x) |v|θ + K ,
Ω
kγk0,∞ sq0
where K =
· meas(Ω) − K1 , and (49) is proved.
The functional k kγ : W01,p (Ω) → R defined by
kvkγ =
µZ
γ(x) |v|
Ω
θ
¶1
θ
W01,p (Ω).
is a norm on
On the finite dimensional subspace X1 the norms k k1,p
f = K(X
f 1 ) > 0 such that
and k kγ being equivalent, there is a constant K
µZ
f
kvk1,p ≤ K
Ω
γ(x) |v|θ
¶1
θ
for all v ∈ X1 .
DIRICHLET PROBLEMS WITH p-LAPLACIAN
375
Consequently, by (49), on X1 it holds:
1 fp
K
F(v) ≤
p
µZ
γ(x) |v|
θ
Ω
¶p
θ
−
Z
γ(x) |v|θ − K
Ω
1 fp
=
K kvkpγ − kvkθγ − K .
p
Therefore
1 fp
K kvkpγ − kvkθγ − K ≥ 0 for all v ∈ S
p
and taking into account θ > p, we conclude that S is bounded.
We also need the following Z2 symmetric version of the “Mountain Pass Theorem” (see e.g. Theorem 9.12 in Rabinowitz [26]).
Theorem 19. Let X be an infinite dimensional real Banach space and let
I ∈ C 1 (X, R) be even, satisfy (PS) condition and I(0) = 0. If:
(I1 ) there are constants ρ, α > 0 such that I|kxk=ρ ≥ α;
(I02 ) for each finite dimensional subspace X1 of X the set {x ∈ X | I(x) ≥ 0}
is bounded,
then I possesses an unbounded sequence of critical values.
Now, we can state
Theorem 20. Suppose the Carathéodory function f : Ω × R → R is odd
in the second argument: f (x, s) = − f (x, −s). If conditions (i), (ii), (iii) of
Theorem 18 are satisfied, that is
(i) there is q ∈ (1, p∗ ) such that
|f (x, s)| ≤ C(|s|q−1 + 1)
for x ∈ Ω, s ∈ R ,
with C ≥ 0 constant;
(ii)
lim sup
s→0
f (x, s)
< λ1
|s|p−2 s
uniformly with x ∈ Ω
where λ1 is the first eigenvalue of −∆p on W01,p (Ω);
(iii) there are constants θ > p and s0 > 0 such that
0 < θ F (x, s) ≤ s f (x, s)
for x ∈ Ω, |s| ≥ s0 ,
then problem (9) has an unbounded sequence of solutions.
376
G. DINCA, P. JEBELEAN and J. MAWHIN
Proof: The function f being odd, the functional F is even. It is obvious
that F(0) = 0.
From (iii) by Theorem 15, F satisfies the (PS) condition.
By (i), (ii) and Theorem 17, there are constants α, ρ > 0 such that
F|kuk1,p =ρ ≥ α.
Proposition 7 and (i), (iii) show that the set {v ∈ X1 | F(u) ≥ 0} is bounded
in W01,p (Ω), whenever X1 is a finite dimensional subspace of W01,p (Ω).
Theorem 19 applies with X = W01,p (Ω) and I = F.
Remark 13. In Proposition 7 by condition (ii), the exponent q in the growth
condition (i) is forced to be in the interval (p, p∗ ) (see Remark 12). Therefore, as
in the case of Theorem 18, the generality of Theorem 20 is not lost if q in (i) is
required to be in the interval (p, p∗ ) instead of (1, p∗ ).
Remark 14. In the particular case p = 2 the symmetry assumption on f allows to remove condition (ii) in Theorem 20 (see e.g. Theorem 9.38 in Rabinowitz
[26], Theorem 6.6 in Struwe [27]).
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378
G. DINCA, P. JEBELEAN and J. MAWHIN
G. Dinca,
University of Bucharest, Faculty of Mathematics,
Str. Academiei 14, 70109, Bucharest – ROMANIA
and
P. Jebelean,
West University of Timisoara, Faculty of Mathematics,
Bv. V. Pârvan 4, 1900 Timisoara – ROMANIA
and
J. Mawhin,
Institut de Mathématique Pure et Appliquée,
Bât. Marc de Hemptinne, Chemin du Cyclotron 2, B-1348, Louvain-la-Neuve – BELGIQUE